US10296708B2 - Computer-implemented method for creating a fermentation model - Google Patents

Computer-implemented method for creating a fermentation model Download PDF

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US10296708B2
US10296708B2 US15/009,903 US201615009903A US10296708B2 US 10296708 B2 US10296708 B2 US 10296708B2 US 201615009903 A US201615009903 A US 201615009903A US 10296708 B2 US10296708 B2 US 10296708B2
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macroreactions
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rates
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implemented method
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Tobias Neymann
Lukas Hebing
Sebastian Engell
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Bayer AG
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    • G—PHYSICS
    • G16—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR SPECIFIC APPLICATION FIELDS
    • G16B—BIOINFORMATICS, i.e. INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR GENETIC OR PROTEIN-RELATED DATA PROCESSING IN COMPUTATIONAL MOLECULAR BIOLOGY
    • G16B5/00—ICT specially adapted for modelling or simulations in systems biology, e.g. gene-regulatory networks, protein interaction networks or metabolic networks
    • G—PHYSICS
    • G06—COMPUTING OR CALCULATING; COUNTING
    • G06F—ELECTRIC DIGITAL DATA PROCESSING
    • G06F17/00—Digital computing or data processing equipment or methods, specially adapted for specific functions
    • G06F17/10—Complex mathematical operations
    • G—PHYSICS
    • G16—INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR SPECIFIC APPLICATION FIELDS
    • G16B—BIOINFORMATICS, i.e. INFORMATION AND COMMUNICATION TECHNOLOGY [ICT] SPECIALLY ADAPTED FOR GENETIC OR PROTEIN-RELATED DATA PROCESSING IN COMPUTATIONAL MOLECULAR BIOLOGY
    • G16B5/00—ICT specially adapted for modelling or simulations in systems biology, e.g. gene-regulatory networks, protein interaction networks or metabolic networks
    • G16B5/30—Dynamic-time models

Definitions

  • the invention relates to a computer-implemented method for creating a model of a bioreaction—especially fermentation or whole-cell catalysis—using an organism.
  • Organism in the sense of the application denotes cultures of plant or animal cells such as mammalian cells, yeasts, bacteria, algae, etc., which are used in bioreactions.
  • a further use is the model-based, optimal process control.
  • This uses a dynamic process model in order to optimize process control with respect to amount of product, product features or formation of by-products or other target quantities in a model-based, predictive closed-loop control circuit.
  • Frahm et al. for a hybridoma cell culture [Fever B, Lane P, Atzert H, Munack A, Hoffmann M, Hass V C, Portner R. 2002. Adaptive, Model-Based Control by the Open-Loop-Feedback-Optimal (OLFO) Controller for the Effective Fed-Batch Cultivation of Hybridoma Cells. Biotechnol. Prog. 18(5):1095-1103].
  • dynamic process models can also be used during process development in order to design experiments with optimal information acquisition.
  • This approach is referred to as model-based design of experiments [Franceschini G, Macchietto S. 2008. Model-based design of experiments for parameter precision: State of the art. Chemical Engineering Science 63(19):4846-4872].
  • said technical use requires a dynamic process model to be already present during the development phase. This should be able to be created as fast as possible from already existing process know-how in order to minimize the time taken for process development.
  • Typical product features in the sense of the application are for example glycosylation patterns of proteins or protein integrity, but are not limited to these.
  • dynamic models used in the above-mentioned context do not have this property.
  • the present approach allows a simple model-based integration of product features.
  • the method described provides in this case one of many possible combinations of elementary modes.
  • the number of macroreactions, and therefore the complexity of the model, is fixed and cannot be altered.
  • the method produces separate models for each process segment. Selection of the kinetics of the individual macroreactions takes into account the stoichiometry of the macroreactions selected.
  • the kinetic parameters (model parameters) are not, however, adjusted to the process data. Instead, the use of separate process segment models generates the changes in the process data. Random selection of the reactions can indeed also be based on a comprehensive database, but the approach described for selection of the kinetics and the selected kinetics cannot represent the course of the process or the behaviour of the organism in the process.
  • the use of several process segment models leads to an unnecessary increase in complexity of the process model.
  • the dependencies, i.e. influences of process quantities or of the process state on process behaviour, are not quantified with this method. Once again, there is no integration of product features. Therefore, this method is not a solution to the above-menti
  • the object was achieved by a method for creating a model of a bioreaction with an organism in a bioreactor, described as follows.
  • the application provides a computer-implemented method for creating a model of a bioreaction—especially fermentation or whole-cell catalysis—with an organism, which comprises the following steps:
  • the process control module communicates on-line with a distributed control system, which is customarily used for controlling the bioreactor.
  • process development modules are used for off-line optimization of the process or for designing experiments.
  • the bioreaction modeling according to the invention is based essentially on the assumption of representative macroreactions, which are a simplified representation of internal metabolic processes. Selection of the reactions requires both biochemical background knowledge, and process know-how.
  • the reactions of the metabolic network, their stoichiometry and reversibility property are input by the user via a user interface or ideally automatically by selecting an organism and its deposited metabolic pathways from a database module, in which the background knowledge about the organism is stored.
  • the metabolic network also called stoichiometric network in the prior art
  • the properties of its individual reactions represent the background knowledge of the organism.
  • the metabolic network contains reactions from metabolic pathways important to the organism, for example glycolysis reactions.
  • the selection contains external reactions.
  • An external reaction in the sense of the application contains at least one component outside of the cell, typically at least one input quantity and/or at least one output quantity (product, by-product, etc.).
  • the metabolic network contains reactions that describe cell growth, e.g. in the form of a simplified reaction of internal metabolites to the external biomass. Without being limited to this, FIG. 5 and Table 1 in the example describe an applicable metabolic network.
  • Each elementary mode is a linear combination of reaction rates from the metabolic pathways—i.e. internal and external reactions of the metabolic network, wherein both the “steady state” condition is fulfilled for internal metabolites and the reversibility or irreversibility of reactions is taken into account.
  • linear combinations of reactions that take into account the “steady state” condition for internal metabolites no internal metabolites can accumulate.
  • An internal reaction in the sense of the application takes place exclusively within the cell.
  • a macroreaction in the sense of the application groups together all reactions that lead from one or more input quantities to one or more output quantities. Each elementary mode therefore describes a macroreaction. Compared to the method of Leifheit et al., in the sense of the application the macroreactions are determined on the basis of the background knowledge entered.
  • the elementary modes (EMs) are combined in a matrix E, preferably in a module for matrix construction, which is configured with a corresponding algorithm.
  • Known algorithms can be used for constructing the elementary modes matrix.
  • METATOOL may be mentioned as an example, without being limited to this: [Pfeiffer T, Montero F, Schuster S. 1999. METATOOL: for studying metabolic networks. Bioinformatics 15(3):251-257.]
  • step b) the (external) stoichiometric matrix N p is used to produce from matrix (E) a matrix consisting of possible macroreactions K.
  • K N p ⁇ E (Formula 1)
  • the column vectors of matrix K describe the macroreactions.
  • the row vectors describe the components of the macroreactions (input and output quantities).
  • the stoichiometry of the macroreactions is entered into matrix K.
  • Each reaction rate that is possible in the sense of the metabolic network can be represented as a positive linear combination of these macroreactions.
  • step c) the available measured data (process know-how) for the bioreaction with the organism are entered.
  • cell count, cell viability, concentrations of substrates, such as carbon sources (e.g. glucose), amino acids or O 2 , products and by-products (e.g. lactate or CO 2 ), process parameters such as temperature and/or pH or product features are determined.
  • This input can be manual by the user or automatic, for example by selecting from a database module for storage of measured data and transferring the selected data into a data analysis module, which is connected to the database module.
  • step d the cell-specific rates of secretion and uptake of substrates and (by-)products—together called specific rates q (t)—and optionally the growth and death rates of the organism ( ⁇ (t), ⁇ d (t)) are calculated in step d).
  • the calculation requires the interpolation of the viable cell count, the total cell count and media concentrations using an interpolation method. Changes of the measured quantities with respect to time can be determined therefrom.
  • the calculated rates q (t), ⁇ (t), ⁇ d (t) provide information about the observable dynamic behaviour of the organism over time.
  • one or more different interpolation method(s) can be used in combination.
  • Leifheit et al. describe determination of the changes of measured quantities with respect to time—e.g. those of the total cell count, those of the viable cell count, or those of other media concentrations—from measured data by means of spline-interpolated measured data [Leifheit, J., Heine, T., Kample, M., & King, R. (2007). Rechnerge Technologye rudautomatische Modelltechnik biotechntreur Rothe [Computer-aided semi-automatic modeling of biotechnology processes]. at— Automatmaschinestechnik, 55(5), 211-218]. This method is hereby incorporated in the application by reference.
  • the above-mentioned rates q (t), ⁇ (t), ⁇ d (t) are calculated from these temporal variations: For example, it is possible to calculate the growth rate of the organism ⁇ (t) from spline-interpolated values of the total cell count X t (t) and of the viable cell count X ⁇ (t) and from the changes of the total cell count with respect to time calculable therefrom
  • the death rate ⁇ d (t) can be calculated, when the course of ⁇ (t) is known, from the course of X ⁇ (t) and the course of the changes of the viable cell count with respect to time
  • the specific rates of another component i q i (t) can be calculated from spline-interpolated values of the viable cell count X ⁇ (t) and of the concentration of the component C i (t) and from the course of the changes with respect to time
  • the measured data from step c) are prepared as follows before the first interpolation:
  • the measured data are displaced (called “shifted” in the application).
  • FIG. 1 shows the preparation/shifting of the measured data in the sense of this application.
  • the prepared data are then used for calculating the changes of the total cell count with respect to time
  • lysis is included in the calculation with a lysis factor K l (Formula 9). This can for example be assumed to be constant over the course of the process.
  • the prepared data are also used for calculating the death rate ⁇ d (t).
  • the possible combinations of specific rates q (t) are presented in a flux-map diagram, for example as in FIG. 2 .
  • This manner of presentation provides a good overview of the calculated specific rates q (t).
  • the contour lines indicate here which regions are physiologically important. If the specific rates q (t) and optionally the further rates ⁇ (t) and ⁇ d (t) have different orders of magnitude and units, these are usually combined by means of simplifications into a specific rate vector ⁇ tilde over ( q ) ⁇ (t) with the same units.
  • the specific rate of a macromolecule that is measured in grams [g] will have the unit
  • composition of this macromolecule is estimated, e.g. based on its C-mol content, its specific rate can be altered from [g] to [C-mol], so that the specific rate has the unit
  • a subset (L) of the EMs is selected on the basis of the data, such that this can represent well the specific rates ⁇ tilde over ( q ) ⁇ (t) from d) and/or the measured data from c) according to a mathematical quality criterion.
  • the number of EMs in the subset (L) should be as small as possible in order to ensure as low a complexity of the process model as possible. However, the subset L should ensure a good description of the process know-how.
  • FIG. 3 shows a representation of the solution space, where the original set of EMs (K) is reduced to a subset (L).
  • step e) the calculated specific rates ⁇ tilde over ( q ) ⁇ (t) and the measured data from c) are usually transferred to a module for selecting the relevant macroreactions, which is configured with corresponding algorithms.
  • step e) i., a data-independent and/or a data-dependent prereduction of matrix K is done in any desired order:
  • the data-independent prereduction is preferably done by means of a geometric reduction. This involves calculating all cosine similarities to all other modes for a randomly selected EM. The EM with the highest similarity is removed from the set. This procedure is repeated until a predefined number of EMs is reached. The desired number is usually defined beforehand for the method. The volume of the solution space can be used as a control variable. It was found that, surprisingly, a distinct reduction in the number of macroreactions while maintaining 90 to 98%, preferably 92 to 95%, of the spanned volume compared to the original volume is possible.
  • the data-dependent prereduction can be done by comparing yield coefficients of the EMs (Y EM ) with the yield coefficients calculated from the specific rates ⁇ tilde over (q) ⁇ (t) from d) (Y m ).
  • the yield coefficient of the k-th EM (Y i,j EM,k ) is determined according to Formula 10 by dividing the corresponding stoichiometric coefficients of the external metabolites i and j. For the k-th EM, these are the matrix entries K i,k and K j,k .
  • the yield coefficient Y i,j m (t) indicates, according to Formula 11, the ratio between two cell-specific rates that have been measured or calculated according to d) ( ⁇ tilde over (q) ⁇ i (t), ⁇ tilde over (q) ⁇ j (t)):
  • the inventive method “linear estimation of reaction rates of selected macroreactions with NNLS”, as described on page 15, for the data-dependent prereduction.
  • the method in the context of prereduction is explained there.
  • the advantage of using the process data in the data-dependent prereduction in the method is that the reduction is process-related and thus more effective and focused.
  • step e) ii. a subset of macroreactions is selected using an algorithm: Selection requires a quality criterion, which makes it possible to quantify how well the specific rates ⁇ tilde over (q) ⁇ (t) from d) and/or the measured data from c) can be represented with a subset (L), and also an algorithm for selecting the subset.
  • the quality criterion used for representing calculated specific rates ⁇ tilde over (q) ⁇ (t) with a subset L can be, according to Soons et al., the sum of squared residuals of the specific rates (SSR q ) according to Formula 12 [Soons, Z. I. T. A., Ferreira, E. C., Rocha, I. (2010). Selection of Elementary Modes for Bioprocess Control. 11th International Symposium on Computer Applications in Biotechnology, Leuven, Belgium, 7-9 Jul. 2010, 156-161].
  • the value for SSR q should be as small as possible.
  • Minimizing SSR q requires determining beforehand the vector r (t i ) for each considered time point t i with the aid of a non-negative least squares algorithm, such that:
  • This method only relates to reversible macroreactions (such as the “free fluxes” mentioned in the source), moreover, dilution effects (hence concentration changes that are not caused by the cells) cannot be taken into account. If the dimensions of the macroreactions and of the measured values do not agree, these measured values cannot be used for estimating reaction rates. This is for example the case when cell growth in the form of formation of external biomass is part of the macroreactions and only the cell dry mass is known from the measured values. In this form, therefore, this method is not suitable for the use of irreversible macroreactions and fed-batch processes.
  • this method can now also be applied to fed-batch processes. Furthermore, by adding a lower limit for the reaction rates of the macroreactions as a boundary condition of the linear optimization problem, the method can also be used for irreversible reactions—like the elementary modes. If the dimensions of the macroreactions and of the measured values are not in agreement, by means of suitable correlations the dimension of the measured values can be adjusted to that of the macroreactions.
  • This combination of the linear estimation according to Leighty et al. with the improvements from this application is designated in the following as “linear estimation of reaction rates of selected macroreactions”.
  • this method also yields an estimated course of the reaction rates r (t) of the subset.
  • the linear optimization problem is solved by using, instead of a linear least squares solver, the non-negative least squares solver (NNLS) by Lawson et al. [Lawson, C. L. and R. J. Hanson, Solving Least Squares Problems, Prentice-Hall, 1974, Chapter 23, p. 161.].
  • NLS non-negative least squares solver
  • the inventive method of “linear estimation of reaction rates of selected macroreactions with NNLS” can be additionally used as a further data-dependent method in relation to prereduction of the EMs in step e) i).
  • a very large set K of macroreactions The result of the method is, firstly, the value for SSR C and, secondly, the course of the reaction rates r (t).
  • EMs with small values of the associated rate r (t) are removed from matrix K. This procedure is repeated until a predefined number of EMs is reached or the value of SSR C exceeds a specified threshold.
  • the set with the smallest minimized SSR q value is selected.
  • random selection is ineffective, as there is a very marked increase in the number of possible combinations. For example, when selecting 10 reactions from a set of 20 000 EMs, there are more than 2.8 ⁇ 10 36 combinations. The probability of finding the optimum combination is then very slight. Owing to the use of the quality criterion SSR q , this method is vulnerable to measurement uncertainties and measurement deviations.
  • Another disadvantage of this method is that a vector ⁇ tilde over (q) ⁇ (t i ) that does not lie within the solution space of all EMs cannot be used. An approximate solution cannot be determined. This is a great disadvantage particularly in the case of uncertain measurements and specific rates. Owing to the use of the quality criterion SSR q , this method is likewise vulnerable to measurement uncertainties and measurement deviations.
  • an evolutionary, especially genetic, algorithm is therefore used for selecting the relevant macroreactions, i.e. for selecting the EM-subset L.
  • Such an algorithm is known for example from Baker et al. [Syed Murtuza Baker, Kai Schallau, Björn H. Junker. 2010. Comparison of different algorithms for simultaneous estimation of multiple parameters in kinetic metabolic models. J. Integrative Bioinformatics:-1-1.1].
  • target function the method “linear estimation of reaction rates of selected macroreactions” is used to calculate the corresponding value SSR C for various combinations of EMs.
  • random selection may also be used.
  • step iii) the validity of the EM-subset L is verified graphically.
  • the flux map from step d) as projection of the EM-subset L.
  • FIG. 4 shows the flux map with the projection of a subset of six EMs. If the EM-subset L is valid, the measured data is still located within the EM-subset L. This representation allows a rapid graphic check of the validity of selection.
  • the specific rates ⁇ tilde over (q) ⁇ (t) from d) and/or the measured data from c) are used to calculate the reaction rates of the macroreactions of the subset L.
  • the calculation of r (t) can be done according to Soons et al. on the basis of the specific rates ⁇ tilde over (q) ⁇ (t) as described in e) [Soons, Z. I. T. A., Ferreira, E. C., Rocha, I. (2010). Selection of Elementary Modes for Bioprocess Control. 11th International Symposium on Computer Applications in Biotechnology, Leuven, Belgium, 7-9 Jul. 2010, 156-161]; preferably, the calculation of r (t) is done on the basis of the measured data from c) using the inventive “linear estimation of reaction rates of selected macroreactions”.
  • step g) of the method the kinetics of the macroreactions are devised.
  • the model parameters to be determined are found from the kinetics.
  • step g) i., the generic kinetics are devised from the stoichiometry of the macroreactions.
  • k a limitation of the Monod type is assumed.
  • the maximum reaction rate is multiplied by the various limitations ⁇ tilde over (r) ⁇ i :
  • the Monod constants K m,k,i and the Hill coefficients n i are the parameters of the equation, which first values are entered manually.
  • the Monod constants K m,k,i are set to a tenth of the respective maximum measured concentrations and the Hill coefficients n i are set to a value of 1.
  • step g) ii., the quantities influencing the reaction rates r (t) determined in f) are determined. This involves considering all quantities which describe the process state (i.e. including bioreaction conditions, for example pH, reactor temperature, partial pressures, which are not derivable from the stoichiometry of the macroreaction).
  • the influencing quantities can be determined manually, for example with the aid of a statistical method such as partial least squares. To this end, the correlation between the process state (which is combined in a matrix) and the reaction rates r (t) from f) is determined.
  • a step g) iii. the influences determined in g) ii. are then quantified and the kinetics from i. are expanded by corresponding terms.
  • the term ⁇ circumflex over (r) ⁇ i is any desired function which, depending on the process state, yields a value between 0 and 1.
  • the generic kinetics of the reaction, as established in g) i., is then multiplied by this term.
  • K l,k,i denotes the inhibition constant and is a further model parameter, which first value is entered manually and is usually set to a tenth of the respective maximum measured concentrations.
  • step h) the values of the model parameters p of the kinetics are adjusted to the reaction rates of the macroreactions r (t) determined in f):
  • model parameter value estimation Numerical solution of one or more differential equations according to Formulas 2 to 4 is not necessary in this step; in independent groups with usually 3 to 10 parameters, the values of the model parameters can be adjusted separately for each macroreaction k. Adjustment is done by a customary method such as the Gauss-Newton method [Bates D M, Watts D G. 1988. Nonlinear regression analysis and its applications. New York: Wiley. xiv, 365.].
  • This model parameter value estimation is especially advantageous for steps i) and j), because on the one hand it can be executed quickly, and on the other hand it provides improved starting values for adjusting the values of the model parameters to measured data from c) in step j).
  • the quality of adjustment is for example calculated using the sum of squared residuals SSR r according to Formula 20:
  • step i) the kinetics of the macroreactions selected in g) is checked for quality of adjustment. This is based on the value SSR r which is calculated in step h) and which quantifies the quality of adjustment of the model parameter value estimation. If the quality of adjustment is unsatisfactory, steps g) and h) may be repeated, until a predefined quality of adjustment is reached.
  • the parameter values of the kinetics from g) can be adjusted to the measured data from c) by a usual method for parameter estimation.
  • the starting values from step h) are preferably used for this adjustment.
  • the model parameter value adjustment takes place with incorporation of the above-mentioned differential equations (Formulas 2 to 4), e.g. by means of the Gauss-Newton method [Bates D M, Watts D G. 1988. Nonlinear regression analysis and its applications. New York: Wiley. xiv, 365.] or using a multiple-shooting algorithm [Peifer M, Timmer J. 2007. Parameter estimation in ordinary differential equations for biochemical processes using the method of multiple shooting. Systems Biology, IET 1(2):78-88].
  • product features can be integrated into the model. Especially preferably, this can be introduced for product features that depend on the concentration of by-products or intermediates. Concentrations of by-products which are external components of the metabolic network entered in a) are already integrated into the model and can be calculated. If required, however, other by-products or intermediates can also be combined in one or more separate metabolic networks. This is advantageous if the expected secretion or uptake rates are of different orders of magnitude or specified metabolic processes are to be examined in different degrees of detail.
  • steps a) to j) it is possible to produce a separate model for calculating the product features, which describes the course of the process of the external components of the separate metabolic network, also with a set of macroreactions with their own kinetics.
  • By-products or intermediates that are not located outside of the organism, but whose accumulation within the cell affects one or more product features, may be externalized in step a) and b) during calculation of the EMs and formulation of the macroreactions, and may thus be classified as external components.
  • the integration of product features that are dependent on concentrations inside the cell or outside the cell may then take place by additional integration of quantitative or qualitative relationships between concentrations and product features.
  • the invention further provides a computer program or software for carrying out the method according to the invention.
  • the model provided using the method according to the invention can be used for process control or planning process control and investigation of the process in the reactor.
  • the background knowledge in the form of a metabolic network was taken from the work of Niu et al. (Metabolic pathway analysis and reduction for mammalian cell cultures—Towards macroscopic modeling. Chemical Engineering Science (2013) 102, pp. 461-473. DOI: 10.1016/j.ces.2013.07.034.).
  • the metabolic network of an animal cell described here contains 35 reactions, which link together 37 internal and external metabolites (see FIG. 5 ; see Table 1).
  • the measured data of the process was taken from Baughman et al., who gives various measured quantities of a fermentation of hybridoma cells over the course of a batch process (cf. FIG. 6 ) [On the dynamic modeling of mammalian cell metabolism and mAb production. In: Computers & Chemical Engineering (2010) 34 (2), pp. 210-222]. The measured data was entered into the method.
  • the C-mol based rate of formation of the antibody can be estimated similarly.
  • the molar composition of the antibody was estimated as CH 1.58 O 0.31 N 0.26 S 0.004 with a formal molar mass of
  • step e an EM-subset of macroreactions was set up, with which the data set was reproduced as well as possible. This required the matrix K from step b). As the number of over 300 000 macroreactions would have led to an excessively large number of possible combinations, a data-dependent prereduction was carried out first.
  • the rates ⁇ tilde over (q) ⁇ (t) determined in step d) were used for calculating the yield coefficients Y m for all combinations of two external metabolites.
  • the lower limit of a yield coefficient Y i,j was selected such that 99% of the determined yield coefficients Y i,j m (t) are above this value.
  • the upper limit was selected such that 99% of the determined yield coefficients Y i,j m (t) are below this value.
  • Table 2 shows some determined limits and also the proportion of EMs having yield coefficients Y i,j EM within these limits. Overall, the number of EMs could thus be reduced to approx. 3000.
  • the reduced matrix K which represents the background knowledge—and the rates ⁇ tilde over (q) ⁇ (t) from d) and the measured data from c)—which form the process know-how—are then used for obtaining a smallest possible subset L of the macroreactions from K.
  • the subset was selected with a genetic algorithm.
  • the linear optimization problem addressed in the “linear estimation of reaction rates of selected macroreactions” was solved.
  • the final sum of the least error squares of the linear optimization problem calculated here was at the same time the value of the target function for the particular selection of the macroreactions.
  • the reaction rates over time were determined.
  • the measured values shown in FIG. 10 were approximated by an estimation of the reaction rates r (t).
  • the result of the method is a piecewise linear course of the individual (volumetric) reaction rates.
  • the cell-specific reaction rates r (t) of the macroreactions shown in Table 3 were obtained.
  • the reaction rates r (t) thus obtained are shown in FIG. 10 .
  • r k,max is the maximum reaction rate
  • N l is the number of limitations taken into account
  • C i is the concentration of the component i
  • K m,k,i are the associated Monod constants
  • n i is the Hill parameter for the reaction order. Their values are adjusted in steps h) and j). Further terms are found from the analysis of the reaction rates r (t) from f). In this example, in addition to substrate limitations, inhibitions according to Formula 26 were also taken into account.
  • the lysis rate K l was assumed to be constant over the process.
  • the parameters C Lac,cr critical lactate concentration
  • ⁇ d,max maximum death rate
  • K d,Lac Monitoring parameter for describing the influence of the lactate concentration on the death rate
  • K l lysis rate introduced by apoptosis and lysis were determined in this step.
  • the course of the estimated concentrations ⁇ (t) was determined from the starting values of the data set, by numerical solution of the ODE system.
  • the difference between the measured concentrations C m (t) and the estimated concentrations ⁇ (t) was minimized by usual methods with the following target function:
  • the model consisting of the matrix L, the kinetics from Table 4 and the kinetics of apoptosis with the associated parameter values from Table 5, was produced as the output.
  • reaction kinetics ⁇ tilde over (r) ⁇ limitation of kinetics r max parameter of reaction kinetics N stoichiometric matrix N p external stoichiometric matrix K matrix that contains macroreactions E matrix that contains all elementary modes X t total cell count X v viable cell count ⁇ growth rate ⁇ d death rate ⁇ tilde over ( ⁇ ) ⁇ growth rate that has been converted from any unit to [ Substance ⁇ ⁇ amount Time ⁇ Cell ⁇ ⁇ count ] K d lysis rate K I parameter of an inhibition limitation K M parameter of a substrate limitation n Hill parameter of an inhibition or substrate limitation L subset of the macroreactions that is used for the model p model parameter S substrate SSR q sum of squared residuals of the specific uptake or secretion rates SSR C sum of squared residuals of the concentration SSR r sum of squared residuals of the reaction rates
  • FIG. 1 shows the shift of measured data: It shows the actual course of a measured quantity (C i (t)), which changes suddenly when there are changes of the dilution rate (D(t)). The course of the shifted concentration (C i,s (t)) only comes from changes caused by the cell.
  • FIG. 2 shows the flux map of two specific rates q 1 and q 2 .
  • the contour lines indicate the frequency with which the particular combination of the rates occurs in the measured data.
  • FIG. 3 shows a three-dimensional representation of the solution space, which is spanned by a positive linear combination of EMs.
  • the solution space of the complete set is shown in black, and that of a subset is shown in grey.
  • FIG. 4 shows the flux map of two specific rates q 1 and q 2 .
  • the 2-dimensional projections of the macroreactions of a set L are shown as vectors.
  • FIG. 5 shows a schematic representation of the metabolic network from Niu et al.
  • the boundary of the cell is shown as a box.
  • the intracellular border of the mitochondrium is shown with a dashed line.
  • External components are marked with the subscript “xt”.
  • the arrows and dotted arrows denote reactions.
  • FIG. 6 shows the measured data of a fermentation with hybridoma cells from Baughman et al.
  • the total cell count (total cells) is calculated here from the total of viable cells and dead cells.
  • the abbreviations GLC, GLN, ASP, ASN, LAC, ALA and PRO denote the substrate glucose and the amino acids glutamine, aspartic acid, asparagine, alanine and proline and the metabolic product lactate.
  • the abbreviation MAB denotes the product of monoclonal antibodies and BM denotes biomass.
  • FIG. 7 shows the growth and death rates and the cell-specific uptake and secretion rates. All cell-specific rates except q MAB are given in
  • FIG. 8 shows the concentrations approximated with the “linear estimation of reaction rates of selected macroreactions” with the selected reaction set.
  • the total cell count (X t ) and the antibody concentration (MAB) were converted for this to C-mol.
  • FIG. 9 shows the minimum error plotted against the number of macroreactions in the subset (n R ).
  • FIG. 10 shows the reaction rates of the macroreactions r (t) determined with the inventive method “linear estimation of reaction rates of selected macroreactions”.
  • FIG. 11 shows the reaction rates of the macroreactions r (t) determined with the inventive method “linear estimation of reaction rates of selected macroreactions” (solid line) together with the algebraically calculated reaction rates ⁇ circumflex over (r) ⁇ ( p , C int (t)) (dashed line).
  • FIG. 12 shows a comparison of the measured concentrations Cm (t) (points) and the simulated process course ⁇ (t) (solid line). The concentrations are given in [mM]. The viable cell count and total cell count (X v /X t in [10 9 cells/l]) and the concentration of the antibody (mAb in [10 ⁇ 4 mM]) are exceptions.

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